Parallel and Consecutive k-out-of-n:F Systems under Dependent Deterioration and Random Stress

In many engineering systems, component failure arises from the interaction between time-varying resistance and random environmental stress. Classical formulations usually assume static resistance and independent components, assumptions often violated in practice. We develop a stochastic framework for parallel and consecutive 𝑘-out-of-𝑛:𝐹 systems in which component resistances follow linear degradation paths with random deterioration rates, while all components are exposed to a common random stress. Dependence among deterioration rates is modelled via a copula, yielding joint lifetime distributions for parallel systems; for consecutive 𝑘-out-of-𝑛:𝐹 structures, maximal signatures express system lifetimes in terms of parallel-system survival. Numerical illustrations with a survival Clayton copula and Weibull marginal deterioration (decreasing, constant, and increasing hazard) examine the impact of dependence strength and marginal failure behaviour on reliability and mean residual life. We also analyse an optimal replacement policy that minimises long-run average cost and compare maintenance decisions under independent and dependent deterioration. The results show that ignoring dependence systematically underestimates failure risk and cost, and that preventive replacement is economically justified only for components with increasing failure rates, whereas constant or decreasing hazard rates favour run-to-failure strategies.